CSCI6268L09 - Foundations of Network and Computer Security...

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Foundations of Network and Foundations of Network and Computer Security Computer Security J J ohn Black Lecture #9 Sep 16 th 2009 CSCI 6268/TLEN 5550, Fall 2009
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Announcements Quiz #2 will be next Friday, Sep 25 th Will cover material up to next Weds 9/23
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Birthday Paradox Need another method Birthday paradox: if we have 23 people in a room, the probability is > 50% that two will share the same birthday This happens because 23 is near the square root of 365 Sqrt(365) ≈ 19.1 More on this in a moment
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Birthday Paradox (cont) Let’s do the math Let n equal number of people in the class Start with n = 1 and count upward Let NBM be the event that there are N o- B irthday- M atches For n=1, Pr[NBM] = 1 For n=2, Pr[NBM] = 1 x 364/365 ≈ .997 For n=3, Pr[NBM] = 1 x 364/365 x 363/365 ≈ .991 For n=22, Pr[NBM] = 1 x … x 344/365 ≈ .524 For n=23, Pr[NBM] = 1 x … x 343/365 ≈ .493 Since the probability of a match is 1 – Pr[NBM] we see that n=23 is the smallest number where the probability exceeds 50%
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Occupancy Problems What does this have to do with hashing? Suppose each hash output is uniform and random on {0,1} n Then it’s as if we’re throwing a ball into one of 2 n bins at random and asking when a bin contains at least 2 balls This is a well-studied area in probability theory called “occupancy problems” It’s well-known that the probability of a collision occurs around the square-root of the number of bins If we have 2 n bins, the square-root is 2 n/2
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Birthday Bounds This means that even a perfect n-bit hash function will start to exhibit collisions when the number of inputs nears 2 n/2 This is known as the “birthday bound” It’s impossible to do better, but quite easy to do worse It is therefore hoped that it takes (2 64 ) work to find collisions in MD5 and (2 80 ) work to find collisions in SHA-1
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The Birthday Bound 1.0 Probability 0.0 0.5 2 n Number of Hash Inputs 2 n/2
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At CRYPTO 2004 (August) Collisions found in HAVAL, RIPEMD, MD4, MD5, and SHA-0 (2 40 operations) Wang, Feng, Lai, Yu Only Lai is well-known HAVAL was known to be bad Dobbertin found collisions in MD4 years ago MD5 news is big! CU team lowered time-to-collision to 3 mins (July 2005)
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CSCI6268L09 - Foundations of Network and Computer Security...

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